Factor-Specific Technology Choice

The purpose of this article is to provide a detailed treatment of a static, two-dimensional problem of optimal factor-specific technology choice. In such a problem, the decision maker faces a menu of local technologies which depend on the quantity of the two factors and their respective quality (i.e., unit productivity). The menu features a trade-off insofar as choosing higher quality of one factor comes at the cost of reducing the quality of the other one. The decision maker is allowed to select her preferred technology, in order to maximize total output/profit/utility, for all configurations of factor quantities. The aggregate function is then constructed as an envelope of local functions. Decision problems with this structure may arise in firms which contemplate not just about the demand for production factors -- such as capital and labor -- but also about the degree of their technological augmentation (Atkinson & Stiglitz 1969, Basu & Weil 1998, Caselli & Coleman 2006). Mathematically equivalent problems are also faced by consumers who are allowed to decide both about the quantity and quality of the demanded goods, as well as by workers (or managers) who allocate their limited endowments of time/effort across two alternative tasks. Hence, despite being motivated primarily by the earlier contributions to the theory of economic growth and factor-augmenting technical change (e.g., Basu & Weil 1998, Acemoglu 2003, Jones 2005, Caselli & Coleman 2006), the appeal of the current paper is in fact much broader. The class of problems which we solve here has applications both in micro- and macroeconomics, and they can be viewed both as producer and consumer problems. Factor-specific technology choice problems of the type studied here are useful, in particular, for addressing issues related to natural resources, human capital and capital--skill complementarity, industrial organization, international trade, labor markets, sectoral change, consumption patterns, social welfare, and so on. Interesting results have already been obtained for certain specific cases of the factor-specific technology choice problem. First, it has been demonstrated that when the technology menu has the Cobb--Douglas form (which may arise, among other cases, if factor-specific ideas are independently Pareto--distributed; Jones 2005) or if the local function is of such form (Growiec 2008a), then the aggregate function also inherits the Cobb--Douglas form. Second, combining a local function of a CES or a minimum (Leontief) form and a CES technology menu yields an aggregate CES function (Growiec 2008b, Matveenko 2010, Growiec 2011,Leon-Ledesma & Satchi 2016). Third, a broader treatment of the properties of factor-specific technology choice problems with a minimum (Leontief) local function, including their intriguing duality properties, has been provided by Rubinov & Glover (1998), Matveenko (1997), Matveenko (2010), Matveenko & Matveenko (2015). While instructive, the minimum function is however an extreme case, particularly problematic when interpreted as a utility function. Fourth, a few promising results for the general factor-specific technology choice problem with an implicitly specified technology menu have also been provided in section 2.3 of Leon-Ledesma & Satchi (2016). Notwithstanding these important special cases, the literature thus far has not devised a general theoretical framework allowing to analyze the factor-specific technology choice problem in its generality. The key contribution of this article is to put forward such a general theory -- one which would frame all these earlier results in a unique encompassing structure. We find that a unique optimal factor-specific technology choice exists for any homothetic local function $F$ and technology menu $G$. Plugging this choice into the local function $F$ leads to a unique homogeneous (constant returns to scale) aggregate function $\Phi$, which may then be transformed to a homothetic form by an arbitrary monotone transformation. We also find that (i) the shape of the aggregate function $\Phi$ depends non-trivially both on $F$ and $G$ unless one of them is of the Cobb--Douglas form, and (ii) the aggregate function $\Phi$ offers more substitution possibilities (i.e., has less curvature) than the local function $F$ unless the optimal technology choice is independent of factor endowments, which happens only if $F$ is Cobb--Douglas or $G$ follows a maximum function. Our second contribution is to construct and solve the dual problem (in a well-defined generalized sense of duality) where, for every technology, the decision maker maximizes output/profit/utility subject to a requirement of producing a predefined quantity with the aggregate technology $\Phi$. Then, by plugging these optimal factor choices into the local function $F$, we obtain the technology menu $G$ as an envelope. The results are fully analogous. At this stage, the duality property also allows us to provide an additional contribution. Namely, we find that in the optimum, partial elasticities of all three objects -- the local function $F$, the technology menu $G$ and the aggregate function $\Phi$ -- are all equal. We then identify a clear-cut, economically interpretable relationship between their curvatures, giving rise to interesting qualitative implications on concavity/convexity and gross complementarity/substitutability along the three functions. The assumption of homotheticity which we make throughout the analysis, while shared by bulk of the associated literature, does not come without costs. The key limitation is due to Bergson's theorem (Burk 1936) which states that every homothetic function that is also additively separable (either directly or after a monotone transformation) must be either of the Cobb--Douglas or CES functional form. Hence, when one wants to go beyond the CES framework, one must either give up homotheticity (e.g., Zhelobodko et al. 2012) or additive separability (e.g., Revankar 1971, Growiec & Muck 2016, this paper). It follows that all the non-CES cases which are covered by the current study but have not been discussed before, cannot be written down as additively separable. We also devote a separate section of the paper to study the link between the technology menu and the distributions of ideas. Indeed, part of the associated literature derives the technology menu as a level curve of a certain joint distribution of ideas (unit factor productivities) where the marginal idea distributions are either independent (Jones 2005, Growiec 2008b) or dependent following a certain copula (Growiec 2008a). Extending these studies, we show that such ``probabilistic'' construction of the technology menu may place a restriction on the considered class of functions $G$, potentially reducing it to the Cobb--Douglas or CES form because of their homotheticity and additive separability (after a monotone transformation). To show this, we adapt Bergson's theorem to the case of copulas, and particularly Archimedean ones.